Serway & Jewett — Energy of a System: Problem 42.

When an object is displaced by an amount \(x\) from stable equilibrium, a restoring force acts on it, tending to return the object to its equilibrium position. The magnitude of the restoring force can be a complicated function of \(x\). In such cases, we can generally imagine the force function \(F(x)\) to be expressed as a power series in \(x\) as \(F(x) = -(k_1x + k_2x^2 + k_3x^3 + \dots)\). The first term here is just Hooke’s law, which describes the force exerted by a simple spring for small displacements. For small excursions from equilibrium, we generally ignore the higher-order terms, but in some cases it may be desirable to keep the second term as well. If we model the restoring force as \(F = -(k_1x + k_2x^2)\), how much work is done on an object in displacing it from \(x = 0\) to \(x = x_{\text{max}}\) by an applied force \(-F\)?